Mr. Trump, according to Mr. Lavrov, “mentioned that certain circles in the U.S. are still exaggerating, although they cannot prove this, the topic of Russia’s interference with the U.S. election.”

Both sides agreed that the two leaders had been eager to pivot from the dispute and discuss other matters, including a new effort to combat cyberthreats and a cease-fire to begin as early as Sunday in a limited area of southwestern Syria.

While such a step in Syria would be small, it appeared to reflect a desire by American and Russian officials to move past their tense flare-ups over the Syria conflict and facilitate a way to end the six-year civil war there. Moscow has been backing the government of President Bashar al-Assad of Syria while the United States has sought to aid opposition groups fighting to oust him.

“I think this is our first indication of the U.S. and Russia being able to work together in Syria,” Mr. Tillerson said.

Trump administration officials had said that Mr. Trump and Mr. Putin would discuss the idea of partnering to establish “safe zones” in Syria as part of his efforts to usher in a new era of cooperation with Moscow.

Announcing it on Friday gave Mr. Trump a tangible achievement coming out of a risky and heavily scrutinized meeting with Mr. Putin, which had been fraught with expectations and suspicion amid the Russia investigations. Members of Congress in both parties had said it would be diplomatic malpractice for Mr. Trump not to broach the issue of Moscow’s election hacking during the session.

And some signaled they were not satisfied even after Mr. Trump exceeded expectations by raising the election interference.

Senator Jack Reed of Rhode Island, the ranking Democrat on the Armed Services Committee, said that Mr. Trump had “capitulated” to Mr. Putin on the issue.

“Putin and Trump may both wish to ‘move forward,’ ” Mr. Reed said, quoting Mr. Tillerson’s account of how the conversation unfolded, “but the American people and the rest of the free world are all saying ‘wait a minute, let’s figure out what happened here and how to protect ourselves from repeat offenses.’ ”

Mr. Putin is known to prepare extensively for international meetings, surrounded by canny career diplomats like Mr. Lavrov, and in close contact with his old associates in the intelligence services. Mr. Trump, by contrast, was given relatively few written briefing materials, aides said, demanded that attendees be limited to two officials per side, and has publicly criticized American intelligence agencies during overseas trips.

White House officials had been determined that Friday’s meeting would not be a repeat of one Mr. Trump held in the Oval Office in May with Mr. Lavrov and Ambassador Sergey I. Kislyak, when he was photographed in images broadcast by Russian state media grinning and clasping hands warmly with the Russian officials. In the end, though, Mr. Tillerson said the two leaders quickly bonded, and their body language indicated as much.

Before they spoke privately, the two presidents smiled, shook hands and praised each other in front of reporters, saying they hoped for a productive relationship.

“President Putin and I have been discussing various things, and I think it’s going very well,” Mr. Trump said, as both men sat spraddle-legged in chairs arranged side by side. They clasped hands and leaned in to listen to each other.

“We look forward to a lot of very positive happenings for Russia, and for the United States, and for everybody concerned. And it’s an honor to be with you,” he told Mr. Putin.

Mr. Putin said he was happy to have the chance to meet Mr. Trump. “We spoke over the phone,” he said, “but phone conversations are never enough, definitely.”

He added: “I hope that, as you have said, our meetings will yield positive results.”

Officials from both sides made sure that was the case. Shortly before the meeting, Mr. Tillerson announced he had named a new envoy, Kurt Volker, to negotiate a resolution to the conflict in Ukraine, where Russia has backed separatists and continued to interfere in violation of a peace agreement. Mr. Putin and Mr. Lavrov discussed the situation in Ukraine, said Mr. Tillerson, who was to travel with Mr. Volker, a former NATO ambassador and official at the National Security Council, to Kiev on Sunday. The State Department said he would hold regular meetings with Ukraine, Russia, Germany and France.

There was no word on whether Mr. Putin asked Mr. Trump to consider lifting the sanctions the United States has imposed on Russia as a result of its activities in Ukraine and annexation of Crimea or the election hacking. There was also no indication of whether he raised the possibility of reclaiming two diplomatic compounds reclaimed by the Obama administration as retaliation for Moscow’s interference in the 2016 contest. Mr. Tillerson did not respond to a reporter’s shouted question on the matter.

But just as the lengthy closed-door talks were winding down, the Trump administration gave word of the cease-fire agreement on Syria, ensuring that the two presidents would have something productive to show for a session that was originally scheduled to last for just 30 minutes.

“There’s so much for us to talk about,” Mr. Tillerson said. “And it was a good start.”